You'll need wire and something to cut the wire. Also useful, something to file the wire and pliers to bend the wire. Additionally, beads or charms can be added.
Most importantly, you need to figure out the circumference of the widest part the bangle will need to slip over: your knuckles. There are two ways to go about this-- find a bangle that fits you well and use that, or take a narrow strip of paper and a piece of tape to make a temporary bangle/template.
Now the fun part begins!
You'll want a sturdy base for your bangle, at least 20 gauge wire, unless you are going to harden the wire or make a lot of wraps as the base to work from. Cut a length 2 or 3 times the length of your circumference plus an inch or two and make circles. Messy wrapped around circles if going for that look, carefully looped if imitating the green or beaded bangle in the pictures.
For the super easy bangles, messy is better. You're going to be adding more wires and randomly wrapping around later, anyway.
Once you have a solid base, the fun continues. Wrap another gauge of wire, another color of wire, whatever you like. Wrap tightly, wrap loosely- you can even wrap through sections of your base like I did with the copper and gold bracelet. Check out some wire weaving tutorials for more on that.
To make the spiraled bracelet, I wrapped wire tightly around a knitting needle and then stretched the "spring" it created out, weaving wire through it.
The green and beaded bangles are the more difficult ones. Tape or clamp to get the wires to line up nicely and then carefully wrap the ends. It is much, much easier to wire wrap like this with flat or half-round wire. Round wire does not make nice wraps (which is why the beaded bangle is messily wrapped).
The variations are just about endless and can be as elegant or as funky as you desire. Just remember to be mindful of wire ends. File them, hide them, make sure they aren't going to leave painful scratches or snag any clothing!